@Article{IJNAM-5-64, author = {Fischer , P. and Tung , K.-K.}, title = {Wavelets, a Numerical Tool for Multiscale Phenomena: From Two Dimensional Turbulence to Atmospheric Data Analysis}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {5}, number = {5}, pages = {64--84}, abstract = {
Multiresolution methods, such as the wavelet decompositions,
are increasingly used in physical applications where multiscale
phenomena occur. We present in this paper two applications illustrating
two different aspects of the wavelet theory.
In the first part of this paper, we recall the bases of the wavelets
theory. We describe how to use the continuous wavelet decomposition for
analyzing multifractal patterns. We also summarize some results about
orthogonal wavelets and wavelet packets decompositions.
In the second part, we show that the wavelet packet filtering can be
successfully used for analyzing two-dimensional turbulent flows. This
technique allows the separation of two structures: the solid rotation
part of the vortices and the remaining mainly composed of vorticity
filaments. These two structures are multiscale and cannot be obtained
through usual filtering methods like Fourier decompositions. The first
structures are responsible for the inverse transfer of energy while the
second ones are responsible for the forward transfer of enstrophy. This
decomposition is performed on numerical simulations of a two dimensional
channel in which an array of cylinders perturb the flow.
In the third part, we use a wavelet-based multifractal approach to
describe qualitatively and quantitatively the complex temporal patterns
of atmospheric data. Time series of geopotential height are used in this
study. The results obtained for the stratosphere and the troposphere
show that the series display two different multifractal behaviors. For
large time scales (several years), the main Hölder exponent for the
stratosphere and the troposphere data are negative indicating the
absence of correlation. For short time scales (from few days to one
year), the stratosphere series present some correlations with Hölder
exponents larger than 0.5, whereas the troposphere data are much less
correlated.